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Population Inferences

Grade 7 · Statistics · Worksheet 1

  1. A city is planning a new park and wants to know what percentage of residents would prefer a swimming pool over a skate park. They survey a random sample of 12,500 residents and find that 7,800 prefer a swimming pool. Based on this sample, how many residents out of the city's total population of 150,000 would be expected to prefer a swimming pool? Answer: ______________
  2. A town has 12,000 residents. Mason surveys a random sample of 250 residents and asks them whether they support building a new community center. In the sample, 127 say yes. Based on this random sample, estimate the number of residents in the entire town who support the new community center. Explain your reasoning. Answer: ______________
  3. A random sample of 135 students from a large school shows that 81 prefer digital textbooks over print. Estimate the number of students in the entire school of 1125 who prefer digital textbooks. Answer: ______________
  4. A random sample of 216 students from a large school shows that 96 prefer year-round school. Estimate the number of students who prefer year-round school in the entire school of 1,296 students. Answer: ______________
  5. Emma surveys a random sample of 75 students at her school about whether they prefer hiking or biking for an outdoor trip. She finds that 45 students prefer hiking. If there are 825 students in the whole school, estimate how many students in the entire school prefer hiking. Justify your answer by showing the proportion from the sample. Answer: ______________
  6. A wildlife conservation team is studying the population of deer in a large forest preserve. They capture and tag 150 deer. Two months later, they capture 200 deer and find that 25 of them have tags. Based on this capture-recapture data, what is the estimated total deer population in the preserve? Answer: ______________
  7. A wildlife biologist is studying the population of deer in a national park. She captures and tags 150 deer. Two weeks later, she captures 200 deer and finds that 25 of them have tags. Based on this capture-recapture data, what is the estimated total deer population in the park? Answer: ______________
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Answer Key & Explanations

Population Inferences · Grade 7 · Worksheet 1

  1. A city is planning a new park and wants to know what percentage of residents would prefer a swimming pool over a skate park. They survey a random sample of 12,500 residents and find that 7,800 prefer a swimming pool. Based on this sample, how many residents out of the city's total population of 150,000 would be expected to prefer a swimming pool? Answer: 93600 Solution: We have a sample of 12,500 residents, and 7,800 of them prefer a swimming pool.
    Full step-by-step solution

    Step 1: Understand the problem We have a sample of 12,500 residents, and 7,800 of them prefer a swimming pool. We want to estimate how many of the total 150,000 residents would prefer a swimming pool, assuming the sample proportion holds for the whole population. Step 2: Find the sample proportion The proportion in the sample who prefer a swimming pool is: Number who prefer pool / Total sample size = 7800 / 12500 Step 3: Simplify the proportion 7800 / 12500 can be simplified by dividing numerator and denominator by 100: 78 / 125 Step 4: Use the proportion for the whole population Multiply the total population by the sample proportion: 150000 × (7800 / 12500) Step 5: Simplify the calculation First, note that 150000 / 12500 = 12 So: 150000 × (7800 / 12500) = 12 × 7800 Step 6: Multiply 12 × 7800 = 93600 Step 7: Conclusion Based on the sample, we would expect 93,600 residents in the city to prefer a swimming pool. Final answer: 93600

  2. A town has 12,000 residents. Mason surveys a random sample of 250 residents and asks them whether they support building a new community center. In the sample, 127 say yes. Based on this random sample, estimate the number of residents in the entire town who support the new community center. Explain your reasoning. Answer: 6096 Solution: Find the proportion of supporters in the sample. 127 out of 250 residents support the center. Since the sample is random, we assume this proportion applies to the entire population of 12,000 residents.
    Full step-by-step solution

    Step 1: Find the proportion of supporters in the sample. 127 out of 250 residents support the center. So the sample proportion is 127 / 250 = 0.508. Step 2: Since the sample is random, we assume this proportion applies to the entire population of 12,000 residents. Step 3: Multiply the population size by the sample proportion: 12,000 × 0.508 = 12,000 × (127 / 250) = (12,000 / 250) × 127 = 48 × 127 = 6,096. Step 4: Therefore, we estimate that about 6,096 residents in the town support building the new community center. The answer is 6096.

  3. A random sample of 135 students from a large school shows that 81 prefer digital textbooks over print. Estimate the number of students in the entire school of 1125 who prefer digital textbooks. Answer: 675 Solution: Find the proportion from the sample: 81 out of 135 students prefer digital textbooks. Proportion = 81/135 = 3/5 = 0.6. Apply this proportion to the total school population of 1125: 0.6 × 1125 = 675.
    Full step-by-step solution

    Step 1: Find the proportion from the sample: 81 out of 135 students prefer digital textbooks. Proportion = 81/135 = 3/5 = 0.6. Step 2: Apply this proportion to the total school population of 1125: 0.6 × 1125 = 675. The estimated number of students who prefer digital textbooks in the entire school is 675.

  4. A random sample of 216 students from a large school shows that 96 prefer year-round school. Estimate the number of students who prefer year-round school in the entire school of 1,296 students. Answer: 576 Solution: Find the proportion from the sample: 96 out of 216 prefer year-round school. Write as a fraction: 96/216. Step 2: Simplify the fraction: divide numerator and denominator by 24 to get 4/9.
    Full step-by-step solution

    Step 1: Find the proportion from the sample: 96 out of 216 prefer year-round school. Write as a fraction: 96/216. Step 2: Simplify the fraction: divide numerator and denominator by 24 to get 4/9. Step 3: Set up a proportion: (4/9) = (x/1296), where x is the estimated number in the whole school. Step 4: Solve for x: multiply both sides by 1296: x = (4/9) * 1296 = (4 * 1296) / 9 = 5184 / 9 = 576. The answer is 576.

  5. Emma surveys a random sample of 75 students at her school about whether they prefer hiking or biking for an outdoor trip. She finds that 45 students prefer hiking. If there are 825 students in the whole school, estimate how many students in the entire school prefer hiking. Justify your answer by showing the proportion from the sample. Answer: 495 Solution: Find the proportion of students in the sample who prefer hiking. 45 out of 75 students prefer hiking, so the proportion is 45/75 = 3/5 = 0.6. Apply this proportion to the entire school population of 825 students.
    Full step-by-step solution

    Step 1: Find the proportion of students in the sample who prefer hiking. 45 out of 75 students prefer hiking, so the proportion is 45/75 = 3/5 = 0.6. Step 2: Apply this proportion to the entire school population of 825 students. Multiply: 0.6 * 825 = 495. Step 3: Justification: Since the sample is random, the proportion in the sample is a reasonable estimate for the proportion in the whole population. Therefore, approximately 495 students in the entire school prefer hiking. The answer is 495.

  6. A wildlife conservation team is studying the population of deer in a large forest preserve. They capture and tag 150 deer. Two months later, they capture 200 deer and find that 25 of them have tags. Based on this capture-recapture data, what is the estimated total deer population in the preserve? Answer: 1200 Solution: - Number of deer initially tagged: 150 - Number of deer captured in second sample: 200 - Number of tagged deer in second sample: 25 Set up the proportion using the capture-recapture formula The proportion of tagged deer in the second sample should equal the proportion of tagged deer in the…
    Full step-by-step solution

    Step 1: Identify the known values from the problem - Number of deer initially tagged: 150 - Number of deer captured in second sample: 200 - Number of tagged deer in second sample: 25 Step 2: Set up the proportion using the capture-recapture formula The proportion of tagged deer in the second sample should equal the proportion of tagged deer in the entire population: 25/200 = 150/total population Step 3: Solve for the total population 25/200 = 150/x Simplify 25/200 = 1/8 So: 1/8 = 150/x Cross multiply: 1 × x = 8 × 150 x = 1200 Step 4: State the final answer The estimated total deer population is 1200.

  7. A wildlife biologist is studying the population of deer in a national park. She captures and tags 150 deer. Two weeks later, she captures 200 deer and finds that 25 of them have tags. Based on this capture-recapture data, what is the estimated total deer population in the park? Answer: 1200 Solution: Set up the proportion using the capture-recapture method formula: tagged in sample / total in sample = tagged in population / total population Plug in the known values: 25 / 200 = 150 / total population Write the equation: 25/200 = 150/x Simplify the left side: 25/200 = 1/8 So 1/8 = 150/x Cross…
    Full step-by-step solution

    Step 1: Set up the proportion using the capture-recapture method formula: tagged in sample / total in sample = tagged in population / total population Step 2: Plug in the known values: 25 / 200 = 150 / total population Step 3: Write the equation: 25/200 = 150/x Step 4: Simplify the left side: 25/200 = 1/8 Step 5: So 1/8 = 150/x Step 6: Cross multiply: 1 * x = 8 * 150 Step 7: Calculate: x = 1200 Step 8: The estimated total deer population is 1200.